Graph algorithms | Planar graphs

FKT algorithm

The FKT algorithm, named after Fisher, Kasteleyn, and Temperley, counts the number of perfect matchings in a planar graph in polynomial time. This same task is #P-complete for general graphs. For matchings that are not required to be perfect, counting them remains #P-complete even for planar graphs. The key idea of the FKT algorithm is to convert the problem into a Pfaffian computation of a skew-symmetric matrix derived from a planar embedding of the graph. The Pfaffian of this matrix is then computed efficiently using standard determinant algorithms. (Wikipedia).

FKT algorithm
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Related pages

Spanning tree | Dual graph | Counting problem (complexity) | If and only if | Perfect matching | Pfaffian orientation | Planar graph | Glossary of graph theory | Tutte matrix | Arthur Cayley | Regular graph | Homeomorphism (graph theory) | Determinant | Adjacency matrix | Complete bipartite graph | Parity of a permutation | Holographic algorithm | Complete graph | Boolean satisfiability problem | Diatomic molecule | Hosoya index | Kuratowski's theorem | Orientation (graph theory) | Partition function (statistical mechanics) | Pfaffian | Lattice graph | Matching (graph theory) | P (complexity) | FP (complexity) | Graph embedding | Skew-symmetric matrix | Domino tiling